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We propose a novel model-based approach for constructing optimal designs with complex blocking structures and network effects for application in agricultural field experiments. The potential interference among treatments applied to different plots is described via a network structure, defined via the adjacency matrix. We consider a field trial run at Rothamsted Research and provide a comparison of optimal designs under various different models, specifically new network designs and the commonly used designs in such situations. It is shown that when there is interference between treatments on neighboring plots, designs incorporating network effects to model this interference are at least as efficient as, and often more efficient than, randomized row–column designs. In general, the advantage of network designs is that we can construct the neighbor structure even for an irregular layout by means of a graph to address the particular characteristics of the experiment. As we demonstrate through the motivating example, failing to account for the network structure when designing the experiment can lead to imprecise estimates of the treatment parameters and invalid conclusions.Supplementary materials accompanying this paper appear online.

Textbooks on reponse surface methodology emphasize the importance of lack-of-fit tests when fitting response surface models and stress that, to be able to test for lack of fit, designed experiments should have replication and allow for pure-error estimation. In this paper, we show how to obtain pure-error estimates and how to carry out a lack-of-fit test when the experiment is not completely randomized, but a blocked experiment, a split-plot experiment, or any other multi-stratum experiment. Our approach to calculating pure-error estimates is based on residual maximum likelihood (REML) estimation of the variance components in a full treatment model (sometimes also referred to as a cell means model). It generalizes the approach suggested by Vining et al. (2005) in the sense that it works for a broader set of designs and for replicates other than center-point replicates. Our lack-of-fit test also generalizes the test proposed by Khuri (1992) for data from blocked experiments because it exploits replicates other than center-point replicates and works for split-plot and other multi-stratum designs as well. We provide analytical expressions for the test statistic and the corresponding degrees of freedom and demonstrate how to perform the lack-of-fit test in the SAS procedure MIXED. We re-analyze several published data sets and discover a few instances in which the usual response surface model exhibits significant lack of fit.

Many chemical and biological experiments involve multiple treatment factors and often it is convenient to fit a non-linear model in these factors. This non-linear model can be mechanistic, empirical or a hybrid of the two. Motivated by experiments in chemical engineering, we focus on 𝐷-optimal designs for multifactor non-linear response surfaces in general. To find and study optimal designs, we first implement conventional point and co-ordinate exchange algorithms. Next, we develop a novel multiphase optimization method to construct 𝐷-optimal designs with improved properties. The benefits of this method are demonstrated by application to two experiments involving non-linear regression models. The designs obtained are shown to be considerably more informative than designs obtained by using traditional design optimality algorithms.

When experiments are performed on social networks, it is difficult to justify the usual assumption of treatment–unit additivity, owing to the connections between actors in the network. We investigate how connections between experimental units affect the design of experiments on those experimental units. Specifically, where we have unstructured treatments, whose effects propagate according to a linear network effects model which we introduce, we show that optimal designs are no longer necessarily balanced; we further demonstrate how experiments which do not take a network effect into account can lead to much higher variance than necessary and/or a large bias. We show the use of this methodology in a very wide range of experiments in agricultural trials, and crossover trials, as well as experiments on connected individuals in a social network.

Many processes in the biological industries are studied using response surface methodology. The use of biological materials, however, means that run‐to‐run variation is typically much greater than that in many experiments in mechanical or chemical engineering and so the designs used require greater replication. The data analysis which is performed may involve some variable selection, as well as fitting polynomial response surface models. This implies that designs should allow the parameters of the model to be estimated nearly orthogonally. A class of three‐level response surface designs is introduced which allows all except the quadratic parameters to be estimated orthogonally, as well as having a number of other useful properties. These subset designs are obtained by using two‐level factorial designs in subsets of the factors, with the other factors being held at their middle level. This allows their properties to be easily explored. Replacing some of the two‐level designs with fractional replicates broadens the class of useful designs, especially with five or more factors, and sometimes incomplete subsets can be used. It is very simple to include a few two‐ and four‐level factors in these designs by excluding subsets with these factors at the middle level. Subset designs can be easily modified to include factors with five or more levels by allowing a different pair of levels to be used in different subsets.

Biochemical mechanism studies often assume statistical models derived from Michaelis–Menten kinetics, which are used to approximate initial reaction rate data given the concentration level of a single substrate. In experiments dealing with industrial applications, however, there are typically a wide range of kinetic profiles where more than one factor is controlled. We focus on optimal design of such experiments requiring the use of multifactor hybrid nonlinear models, which presents a considerable computational challenge. We examine three different candidate models and search for tailor-made D- or weighted-A-optimal designs that can ensure the efficiency of nonlinear least squares estimation. We also study a compound design criterion for discriminating between two candidate models, which we recommend for design of advanced kinetic studies.

One attractive feature of optimum design criteria, such as D- and A-optimality, is that they are directly related to statistically interpretable properties of the designs that are obtained, such as minimizing the volume of a joint confidence region for the parameters. However, the assumed relationships with inferential procedures are valid only if the variance of experimental units is assumed to be known. If the variance is estimated, then the properties of the inferences depend also on the number of degrees of freedom that are available for estimating the error variance. Modified optimality criteria are defined, which correctly reflect the utility of designs with respect to some common types of inference. For fractional factorial and response surface experiments, the designs that are obtained are quite different from those which are optimal under the standard criteria, with many more replicate points required to estimate error. The optimality of these designs assumes that inference is the only purpose of running the experiment, but in practice interpretation of the point estimates of parameters and checking for lack of fit of the treatment model assumed are also usually important. Thus, a compromise between the new criteria and others is likely to be more relevant to many practical situations. Compound criteria are developed, which take account of multiple objectives, and are applied to fractional factorial and response surface experiments. The resulting designs are more similar to standard designs but still have sufficient residual degrees of freedom to allow effective inferences to be carried out. The new procedures developed are applied to three experiments from the food industry to see how the designs used could have been improved and to several illustrative examples. The design optimization is implemented through a simple exchange algorithm.

It is increasingly recognized that many industrial and engineering experiments use split-plot or other multi-stratum structures. Much recent work has concentrated on finding optimum, or near-optimum, designs for estimating the fixed effects parameters in multi-stratum designs. However, often inference, such as hypothesis testing or interval estimation, will also be required and for inference to be unbiased in the presence of model uncertainty requires pure error estimates of the variance components. Most optimal designs provide few, if any, pure error degrees of freedom. Gilmour and Trinca ( 2012 ) introduced design optimality criteria for inference in the context of completely randomized and block designs. Here these criteria are used stratum-by-stratum to obtain multi-stratum designs. It is shown that these designs have better properties for performing inference than standard optimum designs. Compound criteria, which combine the inference criteria with traditional point estimation criteria, are also used and the designs obtained are shown to compromise between point estimation and inference. Designs are obtained for two real split-plot experiments and an illustrative split-split-plot structure. Supplementary materials for this article are available online.

Many industrial experiments involve some factors whose levels are harder to set than others. The best way to deal with these is to plan the experiment carefully as a split-plot, or more generally a multistratum, design. Several different approaches for constructing split-plot type response surface designs have been proposed in the literature since 2001, which has allowed experimenters to make better use of their resources by using more efficient designs than the classical balanced ones. One of these approaches, the stratum-by-stratum strategy has been shown to produce designs that are less efficient than locally D-optimal designs. An improved stratum-by-stratum algorithm is given, which, though more computationally intensive than the old one, makes better use of the advantages of this approach, that is, it can be used for any structure and does not depend on prior estimates of the variance components. This is shown to be almost as good as the locally optimal designs in terms of their own criteria and more robust across a range of criteria. Supplementary materials for this article are available online.

In this article, we present a new method for optimizing designs of experiments for non-linear mixed effects models, where a categorical factor with covariate information is a design variable combined with another design factor. The work is motivated by the need to efficiently design preclinical experiments in enzyme kinetics for a set of Human Liver Microsomes. However, the results are general and can be applied to other experimental situations where the variation in the response due to a categorical factor can be partially accounted for by a covariate. The covariate included in the model explains some systematic variability in a random model parameter. This approach allows better understanding of the population variation as well as estimation of the model parameters with higher precision.